Let $M$ be a $k$ dimensional manifold in $\mathbb R^n$. How to determine if $M$ is contained in some $m$ dimesnional plane ($m\geq k$) ?
I know that one way is to guess a $p\in \mathbb R^n$ and $(n-m)$ vectors $v_1,...,v_{n-m}\in \mathbb R^n$ such that $<x-p,v_i>=0$ for each $x\in\mathbb R^n$ and each $v_i$. But I want a more systematic method. (e.g. I know that for $k=1$, $m=2$, we have to check if $M$ has zero torsion.)
Thanks.
If you have the embedding $f\colon M\to\Bbb R^n$, then you want to look at the various osculating spaces $T^{(k)}_p M\subset \Bbb R^n$. Here $$T^{(k)}_p M = \text{Span}\left(\frac{\partial f}{\partial x^i}\big|_{x(p)},\frac{\partial^2 f}{\partial x^i\partial x^j}\big|_{x(p)},\dots,\frac{\partial^k f}{\partial x^{i_1}\dots\partial x^{i_k}}\big|_{x(p)}\right).$$ (You can check that this is well-defined, independent of local coordinates $x$.) If these stabilize at some stage, i.e., $T^{(k+1)}_pM = T^{(k)}_pM$ for all $p$, then you're done. This will ordinarily happen when the $k^\text{th}$ osculating space is all of $\Bbb R^n$; but if it is a fixed $m$-dimensional subspace as $p$ varies, then $M$ lies in an affine $m$-plane.